Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

Abstract

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$ -type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$ , we show that there are only finitely many polarization types of holomorphic symplectic variety of $K3^{[n]}$ -type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of $0$ -cycles on such varieties.